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Superprimes

I recently came across a type of numbers called as superprimes. A superprime is an integer (such as 7331) such that all its left-to-right initial segments are prime (for 7331 the segments are 7, 73, 733, and 7331, all prime).

The fun fact is, there is a largest possible superprime. Can you find it ?

If I had a proof of the infinitude of emirps I'd just say "There are infinitely many." If instead I said "There must be infinitely many" it would be a statement about how the world should be if there's any justice, rather than a statement of fact :).

Heuristically, one should expect infinitely many emirps: there are about \(O(10^d/d)\) primes with \(d\) digits, so about \(O(10^d/d^2)\) of them would be primes in reverse, as long as there are no unexpectedly negative correlations between primes and reverse primes.

Do you believe there is an infinite string of digits whose initial segments are always prime? (Hint: the two beliefs are equivalent.)

Heuristically, in any base \(b\) one should expect only finitely many superprimes: for any superprime of length \(n\) there are \(b\) possible extensions to length \(n+1\), and only about 1 in \(n \log b\) candidates of that size will be prime. Thus (heuristically) the superprimes will start to dwindle after passing \(n > b/\log b\) digits, until there are no more.

How would anyone even hope to find the largest superprime without either using a computer program or just searching it up? I find that you asking people how they came up with the answer is rather pointless. If anyone writes out a rigorous proof that this is the largest, or even just a solution to arrive at this prime, THEN I will eat my own words.